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Geometriae Dedicata

, Volume 159, Issue 1, pp 185–206 | Cite as

Duality properties of indicatrices of knots

  • Colin AdamsEmail author
  • Dan Collins
  • Katherine Hawkins
  • Charmaine Sia
  • Rob Silversmith
  • Bena Tshishiku
Original Paper
  • 59 Downloads

Abstract

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.

Keywords

Spherical indicatrices Duality Bridge index Superbridge index Stick knots Stick number Spherical polygon 

Mathematics Subject Classification (2000)

53A04 57M25 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Colin Adams
    • 1
    Email author
  • Dan Collins
    • 2
  • Katherine Hawkins
    • 3
  • Charmaine Sia
    • 4
  • Rob Silversmith
    • 5
  • Bena Tshishiku
    • 6
  1. 1.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA
  3. 3.Department of MathematicsEpiscopal High SchoolHoustonUSA
  4. 4.Department of MathematicsHarvard UniversityCambridgeUSA
  5. 5.Department of MathematicsUniversity of MichiganAnn ArborUSA
  6. 6.Department of MathematicsUniversity of ChicagoChicagoUSA

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